Finite Group Databasebeta

← #13179687

Subgroups — #13179687

Order 1728.

3360 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C33×C4×D4C211yesmaximal, characteristic, complemented
2#11723981C211yesmaximal, characteristic, complemented
3#11723979C211yesmaximal, characteristic, complemented
4#11723971C212yesmaximal, complemented
5#11723970C212yesmaximal, complemented
6#11723936C211yesmaximal, characteristic, complemented
7#11723931C211yesmaximal, characteristic, complemented
8#11723925C212yesmaximal, complemented
9#11723924C212yesmaximal, complemented
10C4×C4⋊(C3⋊(C3⋊D3))C211yesmaximal, characteristic, complemented
11#11723913C211yesmaximal, characteristic, complemented
12#10596033—313nomaximal
13C2×C33×D4C2211yescharacteristic
14C22×C33×C4C2212yes—
15C2×#6312C2212yes—
16C2×#6312C2212yes—
17C2×#6312C2212yes—
18C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
19C2×C4⋊(C3⋊(C3⋊D3))C2212yes—
20C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
21C2×C4×C3⋊(C3⋊D3)C2212yes—
22C33×C4⋊C4C2211yescharacteristic
23C33×C22⋊C4C2212yes—
24C33×C42C2211yescharacteristic
25#88864C2212yes—
26#88859C2211yescharacteristic
27#88859C2211yescharacteristic
28#88859C2212yes—
29#88859C2212yes—
30C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
31C4⋊(C3⋊(C3⋊Dic3))C2212yes—
32#88857C2211yescharacteristic
33#88857C2211yescharacteristic
34#88857C2212yes—
35C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
36C32×C4×D4D3113yescomplemented
37#64149—313no—
38#64147—313no—
39#64139—326no—
40#64138—326no—
41#64104—313no—
42#64099—313no—
43#64093—326no—
44#64092—326no—
45C4×C4⋊(C3⋊D3)—313no—
46#64081—313no—
47C23×C33C2312yes—
48C22×C3⋊(C3⋊D3)C2311yescharacteristic
49C22×C3⋊(C3⋊D3)C2312yes—
50C33×D4—22no—
51C2×C33×C4—42no—
52C2×C33×C4C2312yes—
53C2×C33×C4C2311yescharacteristic
54C2×C33×C4C2311yescharacteristic
55C2×C33×C4C2311yescharacteristic
56#6312—42no—
57#6312—42no—
58#6312—42no—
59C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
60C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
61C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
62C2×C3⋊(C3⋊Dic3)C2312yes—
63C4⋊(C3⋊(C3⋊D3))—22no—
64C4⋊(C3⋊(C3⋊D3))—42no—
65C4×C3⋊(C3⋊D3)—22no—
66C4×C3⋊(C3⋊D3)—42no—
67#5562—913no—
68C2×C32×D4C2×D3113yes—
69C22×C32×C4C2×D3126yes—
70C2×#638—326no—
71C2×#638—326no—
72C2×#638—326no—
73C2×C4⋊(C3⋊D3)—313no—
74C2×C4⋊(C3⋊D3)—326no—
75C2×C4×C3⋊D3—313no—
76C2×C4×C3⋊D3—326no—
77C32×C4⋊C4C2×D3113yes—
78C32×C22⋊C4C2×D3126yes—
79C32×C42C2×D3113yes—
80#3757—326no—
81#3752—313no—
82#3752—313no—
83#3752—326no—
84#3752—326no—
85C4⋊(C3⋊Dic3)—313no—
86C4⋊(C3⋊Dic3)—326no—
87#3750—313no—
88#3750—313no—
89#3750—326no—
90C4×C3⋊Dic3—313no—
91C22×C33—22no—
92C22×C33—22no—
93C22×C33—22no—
94C22×C33C2411yescharacteristic
95C2×C3⋊(C3⋊D3)—21no—
96C2×C3⋊(C3⋊D3)—22no—
97C2×C3⋊(C3⋊D3)—22no—
98C2×C3⋊(C3⋊D3)—21no—
99C2×C3⋊(C3⋊D3)—22no—
100C2×C3⋊(C3⋊D3)—21no—
101C33×C4—21no—
102C33×C4—22no—
103C33×C4—21no—
104C33×C4#4112yes—
105C3⋊(C3⋊Dic3)—21no—
106C3⋊(C3⋊Dic3)—22no—
107C3⋊(C3⋊Dic3)—21no—
108C3⋊(C3⋊Dic3)—21no—
109C3×C4×D4C3⋊D3113yescomplemented
110#940—913no—
111#938—913no—
112#930—926no—
113#929—926no—
114#895—913no—
115#890—913no—
116#884—926no—
117#883—926no—
118C4×D12—913no—
119#872—913no—
120C23×C32C22×D3126yes—
121C22×C3⋊D3—313no—
122C22×C3⋊D3—326no—
123C32×D4—226no—
124C2×C32×C4—426no—
125C2×C32×C4C22×D3126yes—
126C2×C32×C4C22×D3113yes—
127C2×C32×C4C22×D3113yes—
128C2×C32×C4C22×D3113yes—
129#638—1226no—
130#638—1226no—
131#638—1226no—
132C2×C3⋊Dic3—313no—
133C2×C3⋊Dic3—313no—
134C2×C3⋊Dic3—313no—
135C2×C3⋊Dic3—326no—
136C4⋊(C3⋊D3)—626no—
137C4⋊(C3⋊D3)—1226no—
138C4×C3⋊D3—626no—
139C4×C3⋊D3—1226no—
140#540—271noSylow
141C2×C33—42no—
142C2×C33C2×#4111yescharacteristic
143C2×C33H(5,𝔽2)11yescharacteristic
144C2×C33H(5,𝔽2)11yescharacteristic
145C3⋊(C3⋊D3)—41no—
146C3⋊(C3⋊D3)—42no—
147C2×C3×D4C2×C3⋊D3113yes—
148C22×C3×C4C2×C3⋊D3126yes—
149C2×#67—926no—
150C2×#67—926no—
151C2×#67—926no—
152C2×D12—913no—
153C2×D12—926no—
154C2×C4×D3—913no—
155C2×C4×D3—926no—
156C3×C4⋊C4C2×C3⋊D3113yes—
157C3×C22⋊C4C2×C3⋊D3126yes—
158C3×C42C2×C3⋊D3113yes—
159#217—926no—
160(C2×D3)⋊C4—913no—
161(C2×D3)⋊C4—913no—
162(C2×D3)⋊C4—926no—
163(C2×D3)⋊C4—926no—
164C4⋊Dic3—913no—
165C4⋊Dic3—926no—
166Dic3⋊C4—913no—
167Dic3⋊C4—913no—
168Dic3⋊C4—926no—
169C4×Dic3—913no—
170C22×C32—226no—
171C22×C32—226no—
172C22×C32—226no—
173C22×C32C23×D3113yes—
174C2×C3⋊D3—613no—
175C2×C3⋊D3—626no—
176C2×C3⋊D3—613no—
177C2×C3⋊D3—626no—
178C2×C3⋊D3—626no—
179C2×C3⋊D3—613no—
180C32×C4—226no—
181C32×C4—213no—
182C32×C4—213no—
183C32×C4#235126yes—
184C3⋊Dic3—613no—
185C3⋊Dic3—613no—
186C3⋊Dic3—626no—
187C3⋊Dic3—613no—
188#127—271no—
189#125—271no—
190#123—272no—
191#123—272no—
192#121—271no—
193#121—272no—
194#121—271no—
195#121—272no—
196C4×D4—271no—
197C4×D4C3⋊(C3⋊D3)11yescharacteristic, complemented
198#117—271no—
199C33#54011yescharacteristic, complemented, Sylow
200C23×C3C22×C3⋊D3126yes—
201C22×D3—913no—
202C22×D3—926no—
203C3×D4—226no—
204C2×C3×C4—426no—
205C2×C3×C4C22×C3⋊D3126yes—
206C2×C3×C4C22×C3⋊D3113yes—
207C2×C3×C4C22×C3⋊D3113yes—
208C2×C3×C4C22×C3⋊D3113yes—
209#67—3626no—
210#67—3626no—
211#67—3626no—
212C2×Dic3—913no—
213C2×Dic3—913no—
214C2×Dic3—913no—
215C2×Dic3—926no—
216D12—1826no—
217D12—3626no—
218C4×D3—1826no—
219C4×D3—3626no—
220C2×C32—426no—
221C2×C32C2×#235113yes—
222C2×C32#1004113yes—
223C2×C32#1009113yes—
224C3⋊D3—1213no—
225C3⋊D3—1226no—
226C2×D4—272no—
227C2×D4—272no—
228C2×D4—272no—
229C2×D4—271no—
230C2×D4—272no—
231C2×D4C2×C3⋊(C3⋊D3)11yescharacteristic
232C22×C4—271no—
233C22×C4—272no—
234C22×C4C2×C3⋊(C3⋊D3)12yes—
235C4⋊C4—271no—
236C4⋊C4—271no—
237C4⋊C4—271no—
238C4⋊C4—272no—
239C4⋊C4—272no—
240C4⋊C4C2×C3⋊(C3⋊D3)11yescharacteristic
241C22⋊C4—272no—
242C22⋊C4—271no—
243C22⋊C4—271no—
244C22⋊C4—272no—
245C22⋊C4—272no—
246C22⋊C4C2×C3⋊(C3⋊D3)12yes—
247C42—271no—
248C42C2×C3⋊(C3⋊D3)11yescharacteristic
249C22×C3—226no—
250C22×C3—226no—
251C22×C3—226no—
252C22×C3C23×C3⋊D3113yes—
253C2×D3—1813no—
254C2×D3—1826no—
255C2×D3—1813no—
256C2×D3—1826no—
257C2×D3—1826no—
258C2×D3—1813no—
259C3×C4—226no—
260C3×C4—213no—
261C3×C4—213no—
262C3×C4#3828126yes—
263Dic3—1813no—
264Dic3—1813no—
265Dic3—1826no—
266Dic3—1813no—
267C32#5562113yescomplemented
268C23—271no—
269C23—272no—
270C23C22×C3⋊(C3⋊D3)12yes—
271D4—22no—
272D4—542no—
273D4—1082no—
274D4—1082no—
275D4—1082no—
276D4—1082no—
277C2×C4—42no—
278C2×C4—271no—
279C2×C4—271no—
280C2×C4—271no—
281C2×C4—272no—
282C2×C4—542no—
283C2×C4—1082no—
284C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
285C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
286C2×C4C22×C3⋊(C3⋊D3)12yes—
287C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
288C2×C3—426no—
289C2×C3C2×#3828113yes—
290C2×C3#64361113yes—
291C2×C3#64366113yes—
292D3—3613no—
293D3—3626no—
294C22—22no—
295C22—22no—
296C22—22no—
297C22—541no—
298C22—542no—
299C22—541no—
300C22—542no—
301C22—542no—
302C22—541no—
303C22C23×C3⋊(C3⋊D3)11yescharacteristic
304C4—21no—
305C4—22no—
306C4—21no—
307C4—541no—
308C4—541no—
309C4—542no—
310C4—541no—
311C4#8907612yes—
312C3#10596033113yescomplemented
313C2—42no—
314C2—1081no—
315C2—1082no—
316C2C2×#8907611yescharacteristic
317C2#1172467911yescharacteristic
318C2#1172468411yescharacteristic
subgroupG/H
center, FrattiniC22C23×C3⋊(C3⋊D3)
commutator, socle, cosocleC22×C33C24
FittingC33×C4×D4C2
hypercenterC4×D4C3⋊(C3⋊D3)
hypocenterC33#540
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C4×D4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C33×C4×D4C2semidirect
2. #11723981C2semidirect
3. #11723979C2semidirect
4. #11723971C2semidirect
5. #11723970C2semidirect
6. #11723936C2semidirect
7. #11723931C2semidirect
8. #11723925C2semidirect
9. #11723924C2semidirect
10. C4×C4⋊(C3⋊(C3⋊D3))C2semidirect
11. #11723913C2semidirect
36. C32×C4×D4D3semidirect
109. C3×C4×D4C3⋊D3semidirect
197. C4×D4C3⋊(C3⋊D3)semidirect
199. C33#540semidirect
267. C32#5562semidirect
312. C3#10596033semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C33×C4×D42
#117239812
#117239792
#117239712
#117239702
#117239362
#117239312
#117239252
#117239242
C4×C4⋊(C3⋊(C3⋊D3))2
#117239132
#105960333